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Landmark-guided elastic shape analysis of human character motions
1. | Florida State University, Department of Mathematics, 208 Love Building, 1017 Academic Way, Tallahassee, FL 32306-4510, USA |
2. | Department of Mathematical Sciences, NTNU -Norwegian University of Science and Technology, 7491 Trondheim, Norway |
Motions of virtual characters in movies or video games are typically generated by recording actors using motion capturing methods. Animations generated this way often need postprocessing, such as improving the periodicity of cyclic animations or generating entirely new motions by interpolation of existing ones. Furthermore, search and classification of recorded motions becomes more and more important as the amount of recorded motion data grows.
In this paper, we will apply methods from shape analysis to the processing of animations. More precisely, we will use the by now classical elastic metric model used in shape matching, and extend it by incorporating additional inexact feature point information, which leads to an improved temporal alignment of different animations.
References:
[1] |
M. F. Abdelkader, W. Abd-Almageed, A. Srivastava and R. Chellappa,
Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455.
doi: 10.1016/j.cviu.2010.10.006. |
[2] |
D. Alekseevsky, A. Kriegl, M. Losik and P. W. Michor,
The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276.
|
[3] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015. Google Scholar |
[4] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9.
doi: 10.5244/C.29.DIFFCV.9. |
[5] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen,
A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.
doi: 10.1137/16M1066282. |
[6] |
M. Bauer, M. Bruveris, S. Marsland and P. W. Michor,
Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.
doi: 10.1016/j.difgeo.2014.04.008. |
[7] |
M. Bauer, M. Bruveris and P. W. Michor,
Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97.
doi: 10.1007/s10851-013-0490-z. |
[8] |
M. Bauer, M. Bruveris and P. W. Michor,
R-transforms for Sobolev H2-metrics on spaces of
plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56.
doi: 10.4310/GIC.2014.v1.n1.a1. |
[9] |
M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436.
doi: 10.1016/j.difgeo.2014.12.008. |
[10] |
A. Bruderlin and L. Williams,
Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104.
doi: 10.1145/218380.218421. |
[11] |
M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp.
doi: 10.1017/fms.2014.19. |
[12] |
Carnegie-Mellon, Carnegie-mellon mocap database, 2003. Google Scholar |
[13] |
E. Celledoni, M. Eslitzbichler and A. Schmeding,
Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.
doi: 10.3934/jgm.2016008. |
[14] |
M. Eslitzbichler,
Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.
doi: 10.1007/s00371-014-1001-y. |
[15] |
M. Fuchs, B. Jüttler, O. Scherzer and H. Yang,
Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102.
doi: 10.1007/s10851-009-0156-z. |
[16] |
L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224. Google Scholar |
[17] |
L. Kovar and M. Gleicher,
Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568.
doi: 10.1145/1186562.1015760. |
[18] |
S. Kurtek, A. Srivastava, E. Klassen and H. Laga,
Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438.
doi: 10.1111/cgf.12063. |
[19] |
H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic,
Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.
doi: 10.1016/j.jtbi.2014.07.036. |
[20] |
W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011. |
[21] |
A. Mennucci, A. Yezzi and G. Sundaramoorthi,
Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445.
doi: 10.4171/IFB/196. |
[22] |
P. W. Michor and D. Mumford,
An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.
doi: 10.1016/j.acha.2006.07.004. |
[23] |
W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15.
doi: 10.1109/CVPR.2004.1315138. |
[24] |
W. Mio, A. Srivastava and S. Joshi,
On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324.
doi: 10.1007/s11263-006-9968-0. |
[25] |
T. Pejsa and I. Pandzic,
State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226.
doi: 10.1111/j.1467-8659.2009.01591.x. |
[26] |
C. Samir, P.-A. Absil, A. Srivastava and E. Klassen,
A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73.
doi: 10.1007/s10208-011-9091-7. |
[27] |
J. Shah,
$H^0$-$
type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.
doi: 10.1090/S0033-569X-07-01084-4. |
[28] |
J. Shah,
An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506.
doi: 10.1016/j.aam.2013.06.003. |
[29] |
E. Sharon and D. Mumford,
2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75.
doi: 10.1109/CVPR.2004.1315185. |
[30] |
A. Srivastava, E. Klassen, S. Joshi and I. Jermyn,
Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428.
doi: 10.1109/TPAMI.2010.184. |
[31] |
J. Su, S. Kurtek, E. Klassen and A. Srivastava,
Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552.
doi: 10.1214/13-AOAS701. |
[32] |
Q. Xie, S. Kurtek, E. Klassen and A. Srivastava,
Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929.
doi: 10.1214/14-EJS938D. |
[33] |
L. Younes,
Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).
doi: 10.1137/S0036139995287685. |
[34] |
L. Younes, P. W. Michor, J. Shah and D. Mumford,
A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.
doi: 10.4171/RLM/506. |
show all references
References:
[1] |
M. F. Abdelkader, W. Abd-Almageed, A. Srivastava and R. Chellappa,
Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455.
doi: 10.1016/j.cviu.2010.10.006. |
[2] |
D. Alekseevsky, A. Kriegl, M. Losik and P. W. Michor,
The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276.
|
[3] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015. Google Scholar |
[4] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9.
doi: 10.5244/C.29.DIFFCV.9. |
[5] |
M. Bauer, M. Bruveris, P. Harms and J. M. Andersen,
A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.
doi: 10.1137/16M1066282. |
[6] |
M. Bauer, M. Bruveris, S. Marsland and P. W. Michor,
Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.
doi: 10.1016/j.difgeo.2014.04.008. |
[7] |
M. Bauer, M. Bruveris and P. W. Michor,
Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97.
doi: 10.1007/s10851-013-0490-z. |
[8] |
M. Bauer, M. Bruveris and P. W. Michor,
R-transforms for Sobolev H2-metrics on spaces of
plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56.
doi: 10.4310/GIC.2014.v1.n1.a1. |
[9] |
M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436.
doi: 10.1016/j.difgeo.2014.12.008. |
[10] |
A. Bruderlin and L. Williams,
Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104.
doi: 10.1145/218380.218421. |
[11] |
M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp.
doi: 10.1017/fms.2014.19. |
[12] |
Carnegie-Mellon, Carnegie-mellon mocap database, 2003. Google Scholar |
[13] |
E. Celledoni, M. Eslitzbichler and A. Schmeding,
Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.
doi: 10.3934/jgm.2016008. |
[14] |
M. Eslitzbichler,
Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.
doi: 10.1007/s00371-014-1001-y. |
[15] |
M. Fuchs, B. Jüttler, O. Scherzer and H. Yang,
Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102.
doi: 10.1007/s10851-009-0156-z. |
[16] |
L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224. Google Scholar |
[17] |
L. Kovar and M. Gleicher,
Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568.
doi: 10.1145/1186562.1015760. |
[18] |
S. Kurtek, A. Srivastava, E. Klassen and H. Laga,
Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438.
doi: 10.1111/cgf.12063. |
[19] |
H. Laga, S. Kurtek, A. Srivastava and S. J. Miklavcic,
Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.
doi: 10.1016/j.jtbi.2014.07.036. |
[20] |
W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011. |
[21] |
A. Mennucci, A. Yezzi and G. Sundaramoorthi,
Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445.
doi: 10.4171/IFB/196. |
[22] |
P. W. Michor and D. Mumford,
An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.
doi: 10.1016/j.acha.2006.07.004. |
[23] |
W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15.
doi: 10.1109/CVPR.2004.1315138. |
[24] |
W. Mio, A. Srivastava and S. Joshi,
On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324.
doi: 10.1007/s11263-006-9968-0. |
[25] |
T. Pejsa and I. Pandzic,
State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226.
doi: 10.1111/j.1467-8659.2009.01591.x. |
[26] |
C. Samir, P.-A. Absil, A. Srivastava and E. Klassen,
A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73.
doi: 10.1007/s10208-011-9091-7. |
[27] |
J. Shah,
$H^0$-$
type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.
doi: 10.1090/S0033-569X-07-01084-4. |
[28] |
J. Shah,
An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506.
doi: 10.1016/j.aam.2013.06.003. |
[29] |
E. Sharon and D. Mumford,
2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75.
doi: 10.1109/CVPR.2004.1315185. |
[30] |
A. Srivastava, E. Klassen, S. Joshi and I. Jermyn,
Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428.
doi: 10.1109/TPAMI.2010.184. |
[31] |
J. Su, S. Kurtek, E. Klassen and A. Srivastava,
Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552.
doi: 10.1214/13-AOAS701. |
[32] |
Q. Xie, S. Kurtek, E. Klassen and A. Srivastava,
Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929.
doi: 10.1214/14-EJS938D. |
[33] |
L. Younes,
Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).
doi: 10.1137/S0036139995287685. |
[34] |
L. Younes, P. W. Michor, J. Shah and D. Mumford,
A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.
doi: 10.4171/RLM/506. |





No. of points | Elastic reparam. | Correct features | Incorrect features |
100 | 17 s (6) | 11 s (3) | 22 s (6) |
200 | 32 s (6) | 22 s (3) | 50 s (6) |
400 | 67 s (6) | 48 s (3) | 95 s (8) |
No. of points | Elastic reparam. | Correct features | Incorrect features |
100 | 17 s (6) | 11 s (3) | 22 s (6) |
200 | 32 s (6) | 22 s (3) | 50 s (6) |
400 | 67 s (6) | 48 s (3) | 95 s (8) |
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